Optimal Sales Schemes for Network Goods∗ Alexei Parakhonyak† and Nick Vikander‡ August 2016
Abstract This paper explores how the sequencing of sales affects consumer behavior in a setting with network effects. A monopolist sets a price for its product and also chooses whether to serve some consumers before others through its choice of sales scheme. We show that the sales scheme that maximizes profits is fully sequential. More broadly, the firm should serve consumers as sequentially as possible, with those in smaller groups often served first. A sequential scheme lets consumers observes previous sales before buying themselves. This allows early buyers to lead by example, and act as opinion leaders for those who follow.
JEL-codes: M31, D42, D82, L12 Key Words: Product launch, Network effects, Sequencing of sales ∗
This paper has benefited from presentations at EARIE 2012, EEA 2015, the University of Edinburgh, the
University of Copenhagen, the Toulouse School of Economics, and from discussions with Maarten Janssen and Andre Veiga. † University of Oxford, Department of Economics. E-mail:
[email protected] ‡ University of Copenhagen, Department of Economics. E-mail:
[email protected]
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1
Introduction
A wide variety of products exhibit network effects, where a consumer’s benefit from buying is increasing in total sales. Network effects can arise from the presence of complementary products (e.g. Tesla cars and electric charging stations1 , handsets and apps for the Apple iPhone, consoles and games for the Sony Playstation), technological compatibility (e.g. operating systems such as Windows, Apple OS, and Chrome Os, or business solutions such as Microsoft Office 365 and Google Apps for Work), or consumer social concerns (for products with a fashion component or in settings where consumption is a social experience).2 Regardless of their source, network effects push consumers to buy products they expect to be popular, and imply that the existing network of users can impact product adoption. In this paper, we consider a monopolist serving a group of consumers, and focus on the following question: how does the impact of network effects between consumers depend on the firm’s sequencing of sales? Specifically, we consider how consumer behavior will depend on whether the firm follows a more sequential product release strategy, relative to a more simultaneous one. Doing so allows us to shed light on which type of release strategy (or ‘sales scheme’) can best exploit network effects to the firm’s own benefit. Our focus on sequencing differs from earlier research on network effects, which has instead looked at how firms can exploit network effects through pricing, advertising, seeding strategy, release of limited-time or lower-quality product versions, and technological compatibility. One interpretation of sequential versus simultaneous sales is explicitly in terms of timing. 1
Li et al. (2015) estimate network effects in the electric vehicle market and find them to be substantial,
on both sides of the market. Drivers are more willing to buy electric cars if there are many public charging stations, and investors are more likely to build charging stations if there are many drivers with electric cars. 2 Customers may prefer attending nightclubs or watching movies alongside others, and purchasing clothes, books or music may associate the owner with the ‘in thing’, facilitating social interactions. For products such as multiplayer online games (from PlayStation, Xbox and others) these social interactions can be crucial in determining consumer payoffs.
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A firm looking to serve multiple markets can release its product to all markets simultaneously, or to some markets before others. For example, Tesla announced that when launching its Model 3 sedan in 2017, it will first begin deliveries on the West Coast of the United States, then move east, and then move on to Europe and other markets.3 Sony also considered whether or not to follow a sequential release when launching the PlayStation 4.4 In a similar spirit, when Google provides its Chromebook laptop, with Chrome OS and Google Apps, to different school districts, it can decide whether to approach some districts before others. In business-to-business transactions, where firms approach customers on an individual basis, they can decide explicitly in which order to do so. An alternative interpretation of sequential versus simultaneous sales is in terms of information release. By committing to release pre-order information for their products, firms can effectively make consumer decisions sequential, from a strategic point of view, regardless of whether products are actually delivered to different consumers at different times. For example, Tesla systematically released pre-order sales information for its Model 3, and these widely-reported figures revealed how many customers had signed up for the new model.5 Similarly, when Apple launched the iPhone 5, it publicized pre-order sales figures prior to the official release.6 In a similar spirit, a restaurant can select a layout that helps passers-by see how many patrons are already inside, and a nightclub can place a conspicuous queue outside its entrance, both of which affect how much information potential consumers receive about the number of earlier arrivals.7 3
See “Tesla Model 3 Reservations FAQ,” www.tesla.com/en CA/support/model-3-reservations-faq, ac-
cessed August 8, 2016. 4 According to Sony’s European Marketing director: “We will launch [the PlayStation 4] this year. Exactly what regions, what timing, is being worked through. Which regions in 2013 - is it all of them, is it some of them? Is there some degree of phasing? We’ll reveal that in more detail later, but we can’t yet.” See “PlayStation 4 launch in ‘at least’ one country in 2013”, digitalspy.co.uk, February 22, 2013. 5 See “The Week that Electric Vehicles Went Mainstream,” www.tesla.com/en CA/blog, April 7, 2016. 6 See “iPhone 5 Pre-Orders Top Two Million in First 24 Hours,” Apple.com, September 17, 2012. 7 This interpretation of sequential versus simultaneous sales also relates broadly to the literature on
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Intuition suggests that sequencing can matter for sales dynamics. Given a sequential sales scheme, with either of the above interpretations, consumers can observe the decisions of previous buyers, so that success can breed success. Consumers who observe high sales will increase their own demand, both directly due to the observed installed base, and indirectly as they expect high sales in the future. But by the same token, failure can also breed failure, where low initial sales dissuade other consumers from buying. For this reason, it is not a priori clear what type of sequencing will allow a firm to best exploit network effects to increase expected profits. To investigate this issue, we consider a setting where a firm sells a homogeneous good, and where each consumer’s payoff from buying is the sum of two terms: an intrinsic payoff, and a network payoff that is increasing in total sales. Both a consumer’s intrinsic payoff from buying and the weight he places on the network payoff are private information.8 The firm sets a price and chooses a sales scheme, which is a partition of consumers into different cohorts. Consumers in each cohort buy simultaneously, but consumers in different cohorts buy sequentially, having observed sales from all previous cohorts. The set of all potential sales schemes represents all possible ways of sequencing sales, from fully sequential (one consumer per cohort), to fully simultaneous (all consumers in one cohort). We show which sales scheme is optimal, and more broadly, rank a wide variety of schemes in terms of their expected profits. Our first set of results considers simultaneous versus sequential sales. We show that the optimal scheme is fully sequential, with a single consumer per cohort. The intuition behind the result is as follows. An early consumer realizes that those arriving later will observe her purchase decision, and therefore takes into account how her own purchase will encourage Bayesian Persuasion (see, e.g. Rayo and Segal 2010, Kamenica and Gentzkow 2011). There, a Sender typically commits to a signal structure, which in turn determines the information available to the Receiver. 8 As discussed in Section 3, all of our results continue to hold if the weight consumers place on the network payoff is public information.
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others to buy themselves. This expectation of high later sales increase the consumer’s own expected payoff from buying. Thus, while early failure can encourage later failure under a sequential scheme, the use of this scheme itself makes early failure less likely by encouraging early consumers to buy. The important point is not just that consumers are observed but that being observed makes consumers behave differently. In practice, a fully sequential scheme will likely be feasible in some settings (e.g. sale of bespoke products, or business-to-business transactions), but not in others. This may be due to institutional constraints that limit the set of feasible sales schemes. For example, a firm may find it feasible to release a mass-market product sequentially across different markets, but not sequentially to different consumers within each market. For a firm with capacity constraints, it may be infeasible to release its product simultaneously to very many consumers, if these constraints limit the speed at which the firm can produce additional units. In such settings, it may be these constraints, rather than any strategic considerations, that lead a firm to avoid certain sales schemes. We address this issue by showing that if only a subset of potential sales schemes are feasible, then the scheme that yields the highest expected profits is the one that is most sequential.9 We also show that the scheme which yields the lowest profits is fully simultaneous. Thus, for a firm with at least some control over sequencing, our mechanism suggests avoiding a fully simultaneous sales scheme, and instead opting for a scheme that is as sequential as possible (i.e. from the set of feasible schemes). These results on sequential sales hold regardless of whether a firm is able to commit to its choice of sales scheme, so even if it can change to a different scheme after observing early sales. With the benefit of hindsight, a firm with low early sales may realize that a fully simultaneous scheme would have led to higher realized profits. But from then on, it will still prefer to serve the remaining consumers with a scheme that is as sequential as possible. 9
Put another way, the more sequential the sales scheme (in a way that we make precise), the stronger is
the strategic effect that pushes consumers to buy via network effects.
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Our second set of results looks specifically at the ordering of sales.10 Given a group of sales schemes that are equally sequential, in the sense of having the same number of cohorts of each different size, the most profitable scheme is the one that serves consumers in smaller cohorts first. This result suggest that when a firm launches its product across different markets, the best way to exploit network effects is to start with smaller markets before moving on to larger ones. We also extend the analysis by assuming the firm can partially distinguish between different consumers, specifically when the weight consumers place on the network payoff is public information. The optimal sales scheme then serves consumers sequentially in increasing order of these weights, so that independent-minded consumers make purchase decisions first and can serve as opinion leaders for those who follow. Moreover, we discuss how this result on ordering relates to the broader economics literature on leadership. We then derive an additional result that makes the link between sequential sales and consumers communication. It shows that if consumers can exchange messages about their respective valuations before making purchases, then truthful communication can be incentive compatible, even if the firm may exploit this information to raise the price. The implication is that the sequencing of sales will be less important for products about which consumers regularly communicate. The caveat is that consumer concern that the firm is monitoring their messages can potentially derail successful communication. The vast literature on network goods, starting from seminal papers by Katz and Shapiro (1985) and Farrell and Saloner (1985, 1986), has largely assumed that the order of consumer entry is predetermined, or endogenously chosen by consumers themselves. In contrast, our framework captures the idea that firms may have (at least some) control over the sequencing of sales. Although the main strategic concern of this paper has been pointed out in the literature11 , ours is the first paper to examine how a monopolist can exploit this concern by 10
These results are relevant under the first interpretation of sequential versus simultaneous sales described
above, i.e. explicitly in terms of timing. 11 See Ochs and Park (2010): “A dynamic adoption process, however, introduces a strategic consideration
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sequencing sales, in a setting with rational, forward-looking consumers. Dou et al. (2011) considers a monopolist’s decision to divide consumers into segments and release its product to one segment before another. However, they assume consumers are myopic, whereas consumer expectations of future sales lie at the heart of our mechanism. Many papers analyzing network effects exhibit multiple equilibria, as consumers have self-fulfilling beliefs on how many others are going to enter (see, e.g. Dybvig and Splatt 1983, Cabral et al. 1999). We avoid a problem of multiple equilibria by assuming that (i) the monopolist has control over the timing of potential entry and (ii) there are “extreme” types, whose decisions do not depend on their beliefs about other consumers’ purchase behavior. More broadly, earlier research has looked at a variety of ways that a firm can exploit network effects by adjusting different marketing variables. These include price and advertising (Kalish 1985, Dhebar and Oren 1985, Dockner and Jorgensen 1988, Xie and Sirbu 1995), introduction of complementary goods (Basu et al., 2003), and release of clone products (Sun et al., 2004). Certain results in Padmanabhan et al. (1997) touch on sequencing, showing that a firm may want to first release a product to experts followed by a lower quality version for novices. However, their analysis focuses on how sequential quality provision can help the firm signal private information about the strength of network effects, something which plays no role in our setting. We view our results as complementing those in the literature on network effects, by exploring a novel mechanism through which the firm can exploit these effects, via the sequencing of sales. Our paper also relates to a small literature on optimal sequencing that has not considered network effects, but has instead assumed that consumers have private information about product quality. Typically, in this literature, a firm serves consumers simultaneously or sequentially, consumers can learn from observing previous sales, and they make one-off, that is absent in the static game. Individuals who chose to enter early may influence the entry decisions of others who have not yet entered. This creates the possibility that early entrants may launch a domino chain reaction of widespread adoption,” (p.690).
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irreversible purchase decisions. Sgroi (2002) shows that simultaneously serving a group of ‘guinea pigs’ can prevent an information cascade where all later consumers refrain from buying. Liu and Schiraldi (2012) show that the optimal scheme is often fully simultaneous when prior beliefs are low. Bhalla (2013) suggests instead using simultaneous sales when the firm’s updated beliefs about quality are high, if it can adjust its price over time. Aoyagi (2010) argues that a seller should use sequential sales as a means to implement dynamic pricing.12 Our results complement those in this literature by showing how sequencing affects expected profits in a different economic setting, i.e. one with network effects, rather than quality uncertainty.13 Of these papers, Aoyagi (2010) is the closest to ours in terms of results, showing the optimality of sequential sales and targeting more independent consumers first.14 However, the economic mechanism driving our results is very different. In Aoyagi (2010), the payoff from buying depends directly on the signals received by other consumers, but not on their actual purchase decisions. Consumers there observe whether others have bought the product and use this information to help infer these consumers’ signals. As in the above-cited work on quality uncertainty, there are no consumption externalities, and no strategic interactions between consumers. In contrast, our mechanism relies crucially on strategic interactions, as forward-looking consumers try to influence each others’ behavior. 12
These papers relate to a broader literature on how firms can influence social learning, through means
such as pricing or product testing (see, e.g., Ottaviani and Prat 2001, Bar-Isaac 2003, Bose et al. 2006, Bose et al. 2008, Gill and Sgroi 2008, Gill and Sgroi 2012). 13 In certain settings, sequencing might plausibly affect expected profits via multiple channels. For example, sequencing could affect both the extent to which consumers learn from each other about product quality (as in the literature on quality uncertainty) and the extent to which network effects influence consumer behavior (as analyzed in this paper). Intuitively, the size of the network effects and the degree of quality uncertainty should influence the relative importance of these two channels. 14 Our analysis also differs from Aoyagi (2010) in suggesting what sales scheme to use when a firm has incomplete control over sequencing, in particular whether to serve smaller or larger cohorts first, and in making the link with consumer communication.
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The literature on dynamic platform competition (see recent papers by Cabral 2011, Halaburda et al. 2015) looks at strategic considerations faced by firms in the presence of network effects, and generally focuses on pricing. Veiga (2015) considers a monopolistic platform in continuous time and, as the aforementioned papers, examines the trade-off between attracting new consumers and exploiting existing ones. Although this literature looks at the dynamic sales problem in the presence of network effects, as our paper does, to the best of our knowledge no earlier work considers firm control over the timing of sales. Thus, our paper adds the timing or sequencing dimension to the well-know price dimension from the analysis of dynamic platforms. The rest of the paper is organized as follows. Section 2 presents the model. Section 3 contains the main analysis, looking in turn at simultaneous versus sequential schemes, the ordering of sales, and the connection with direct consumer communication. Section 4 discusses issues of robustness, and Section 5 then concludes. Proofs of all Propositions and of technical lemmas can be found in the appendix.
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Model
A seller of a network good faces a market of n consumers who each have unit demand. Consumers differ in their type (θ, λ), where subscript i denotes the type of consumer i. Both dimensions of type are drawn independently, θi from a uniform distribution U ∼ [θ, θ] and λi from a distribution F on (0, λ), with λ ≡ θ − v0 . We assume for the main analysis that both dimensions of type are private information, but later relax this assumption to explore the situation where values of λ are publicly known. If consumer i buys a unit of the good, then his payoff consists of an intrinsic and a network component, θi +
λi X xj − p, n − 1 j6=i
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(1)
where xj = 1 if consumer j buys and xj = 0 if he does not, and where p is the price. The consumer’s intrinsic payoff from buying is θi , and his network payoff from buying is proportional to the number of other consumers who buy. Thus, λi captures the weight consumer i places on this network effect, or equivalently his sensitivity to other consumers’ purchases. If consumer i does not buy, then he obtains payoff v0 from his outside option, where v0 < θ. This constraint on v0 implies that consumers with a sufficiently high intrinsic payoff will always choose to buy if the seller sets a sufficiently low price. The timing of the game is as follows. At t = 0, the seller sets a price p ∈ R≥0 and selects a sales scheme. The sales scheme determines the extent to which consumers buy simultaneously or sequentially. Specifically, the seller chooses a number of cohorts m ≤ n and how to partition the n consumers between the m different cohorts, I = {I1 , . . . , Im }. We do not restrict a priori the set of all possible sales schemes. However, because type is unobservable, the seller cannot distinguish between different consumers. This means that the seller’s choice of sales scheme I is effectively a choice of m (the number of cohorts) and the cardinality of I1 , . . . , Im (the size of each cohort). At t = 1, all consumers in cohort I1 simultaneously decide whether to buy a unit of the good. Similarly, for any period t with 2 ≤ t ≤ m, all consumers in cohort It simultaneously decide whether to buy, having observed the choice of consumers in all previous cohorts It0 for t0 ≤ t − 1. After consumers in cohort Im make their purchase decisions, payoffs accrue to all players, and the game ends. There is no discounting. For the main analysis, we assume static pricing, where the seller fixes p at t = 0.15 Moreover, we assume that the seller commits at t = 0 to its choice of sales scheme. We later show that our conclusions remain unchanged if the seller is unable to commit to its chosen scheme, and also address the issue of dynamic pricing. We also assume, as in the literature on sequencing with quality uncertainty, that con15
One possible reason for static pricing is consumer fairness concerns, in the sense that consumers may
consider price changes to be unfair. For further discussion, see Dou et al. (2013) and the references therein.
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sumers’ purchase decisions are irreversible. An interpretation of irreversibility is that a consumer has an urgent need for the good or for a suitable alternative. The consumer either buys the good from the seller or exits the market by purchasing a default option, which gives a payoff of v0 , without the possibility to reenter the market in the short run. Another interpretation is that irreversible purchase decisions may be supported by an “exploding offers” strategy employed by firms (see Armstrong and Zhou 2015).16 To describe the strategy of a consumer i in cohort It , note that the relevant history is the number of consumers in cohorts I1 , . . . , It−1 who bought the good. Denote this number by Kt . For a given sales scheme I such that i ∈ It , and a given price p, the strategy of consumer i is a decision rule that, for any Kt , specifies whether or not to buy, xi = 0 or xi = 1. The seller’s strategy is a choice of p and I. We look for a perfect bayesian equilibrium where the strategy of consumer i maximizes his expected payoff, for any history Kt . All expectations follow from Bayes’ rule and other consumers’ equilibrium strategies.17 The seller’s strategy maximizes expected P profits, p 1≤i≤n E(xi ), where we focus on ranking different sales schemes and solving for the optimal I. We assume θ + 1 < v0 < θ to guarantee interior solutions as described below.
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Analysis
To begin the analysis, we take some preliminary steps to describe consumers’ incentives. Consider some consumer i who must decide whether or not to buy after observing sales from 16
This assumption is consistent with our desire to model situations where the seller has at least some control
sequencing. Making the alternative assumption that purchase decisions are reversible would potentially expose us to the problem of multiple equilibria, as in Ochs and Park (2010), which would significantly complicate our analysis (see the discussion following Proposition 1). 17 After any particular history, seller and consumer beliefs about the type of consumers who have yet to act are always given by the prior. Thus, our solution will closely resemble a subgame perfect equilibrium, where the role of unobservable type is to generate demand uncertainty.
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previous cohorts. Suppose that consumer i is in cohort It , that he observes Kt previous sales, and that the price is p. Let Nt denote the number of consumers who will buy in his own cohort It , and let Nt0 denote the number of consumers who will buy in a later cohort It0 . Neither Nt nor Nt0 are known to consumer i, so his purchase decision will depend on how he expects other consumers to behave. Consumer i will find it optimal to buy himself if λi θi + n−1
! X
Kt + E(Nt − xi |Kt ) +
E(Nt0 |Kt , xi = 1)
− p ≥ v0 .
(2)
t+1≤t0 ≤m
The left-hand-side of (2) gives consumer i’s expected payoff from buying, which follows from (1). It consists of the intrinsic payoff from buying, θi , minus the price, plus the expected network payoff, which depends on three components: previous sales, Kt , expected sales from the current cohort, and expected sales from later cohorts. Consumer i’s observation of previous sales will affect both his own behavior and the number of other consumers he expects to buy. By the same reasoning, all consumers in later cohorts will also observe whether consumer i chose to buy before making their own choice, which means that consumer i’s action can influence their behavior. This is why the final expectation in (2) is conditional on consumer i’s decision to buy, xi = 1. Consumer i will find buying optimal if the left-hand-side of (2) exceeds v0 , the payoff from his outside option. Expression (2) shows that the incentive for any consumer i to buy is increasing in his intrinsic payoff from buying. Substituting expected demand from each cohort into this expression and rearranging, the best response of consumer i ∈ It after history Kt is to buy if and only if θ ∈ [θi∗ , θ], where θi∗ (λi ) = v0 + p −
λi Kt + n−1
X
E(xj |Kt ) +
X X
E(xj |Kt , xi = 1) .
(3)
t0 ≥t+1 j∈It0
j∈It \{i}
Consumer i uses a cut-off strategy, in the sense that he buys if θ exceeds a threshold value given by the right-hand-side of (3). This cutoff depends on the particular history he observes and on his value of λ. The consumer with θ = θi∗ (λi ) earns exactly v0 from buying which 12
leaves him indifferent with his outside option. The probability that consumer i will buy after history Kt , from the perspective of those who observe the history but are uncertain about his type, is E(xi |Kt ) =
θ − θi∗ , θ−θ
(4)
where θi∗ ≡ Eλ (θi∗ (λ)) is the expectation of (3) taken with respect to λ. We now verify that 0 < E(xi |Kt ) < 1. This means that we have interior solutions where the probability of buying is always strictly positive but also strictly less than one. The parameter assumptions θ+1 < v0 < θ combined with (3) directly ensure that E(xi |Kt ) < 1. To see that E(xi |Kt ) > 0, the firm’s optimal choice of p is bounded above by the price it would charge a consumer following the best possible history, where all other consumers have bought, who therefore has the highest possible willingness to pay. From (3) and (4), expected profits from this consumer are θ−v0 −p+E(λ) , where λ < λ implies p, yielding optimal price p∗ = θ−v0 +E(λ) 2 θ−θ p∗ < θ − v0 . The optimal price after any other history therefore satisfies p ≤ p∗ < θ − v0 , where (3) and (4) then yield E(xi |Kt ) > 0. From θi∗ ≡ Eλ (θi∗ (λ)) and (3), write X X X E(λ) θi∗ = u0 − Kt + E(xj |Kt ) + E(xj |Kt , xi = 1) , n−1 0 t0 ≥t+1 j∈It \{i}
(5)
j∈It
where u0 ≡ v0 + p denotes a consumer’s effective outside option, taking into account the price. Once again (4) and (5) imply 0 < E(xi |Kt ) < 1. From an ex ante perspective, the overall probability that consumer i will buy depends on his probability of buying after a particular history Kt and on the ex ante probability distribution over all possible histories. Our assumption that θ is uniformly distributed reduces the problem from analysing the whole distribution of relevant histories to just the expected number of consumers who will buy, E(Kt ). This assumption makes the analysis tractable, and combined with λi ≤ λ, allows us to establish equilibrium existence and uniqueness. 13
Proposition 1. For any sales scheme I, the game has a unique perfect bayesian equilibrium. That is, for any consumer i ∈ It and history Kt , the cut-off function θi∗ (λi ) is uniquely defined. The fact that every seller strategy yields a unique value for expected profits is useful when comparing different schemes. Since the equilibrium strategy profile of consumers is unique, given any choice of price and sales scheme, the seller can unambiguously rank different schemes based on their expected profits. If multiple equilibrium consumer strategy profiles were consistent with a single scheme, then multiple values of expected profits would be possible. The ranking of different schemes could then be ambiguous and depend on factors outside of the formal modeling framework, which might influence equilibrium selection.
3.1
Sequential versus simultaneous sales
Given uniqueness, we are now in a position to examine now the impact of network effects on consumer behavior depend on the firm’s sequencing of sales. In particular, we are interested in whether the seller benefits most from using a simultaneous or a sequential scheme. In order to do so we make the following definition. Definition 1. A sales scheme I’ = {I10 , . . . , It0 , . . . , Im0 } is more sequential than another scheme I = {I1 , . . . , It , . . . , Im } 6= I’ if any two consumers in the same cohort under I’ are also in the same cohort under I: i ∈ It0 and j ∈ It0 implies i ∈ It and j ∈ It , for some t ∈ {1, . . . , m}. We say that a first sales scheme is more sequential than a second one if all consumers who were served sequentially and at least some who were served simultaneously in the second scheme are served sequentially in the first. This is equivalent to saying that the second scheme can be transformed into the first by repeatedly breaking up cohorts, taking groups of consumers who were served simultaneously and instead serving some of these consumers be14
fore others. Alternatively, the first scheme can be transformed into the second by repeatedly combining together cohorts that, loosely put, lie next to one another. Definition 1 allows us to make pairwise comparisons between many sales schemes in an intuitive way. Applying the definition, every possible scheme is more sequential than a scheme with all consumers in one cohort (fully simultaneous), and a scheme with a single consumer per cohort (fully sequential) is more sequential than every other possible scheme. The following result says that the seller can best exploit network effects by using a sales scheme that is as sequential as possible. Proposition 2. Suppose that a sales scheme I’ is more sequential than another scheme I, according to Definition 1. Then I’ delivers strictly higher expected profits. This result supporting sequential sales is relevant for a variety of situations with imperfect control over sequencing. For example, when a firm launches its product across M markets, say that have the same size, it may be feasible to serve consumers sequentially between different markets, but not sequentially within a given market. The set of feasible sales schemes is then the set of partitions that place all consumers Ni in market Mi in the same cohort, for each i = 1, . . . , M . Thus, a number of feasible schemes have strictly fewer than M cohorts, and they correspond to simultaneously launching the product to different subsets of the set of M markets. There is also a single feasible scheme with exactly M cohorts, which corresponds to product launch in each market, one after the other. This last scheme is as sequential as possible, from the set of feasible schemes, and will therefore be most profitable in our setting. Interpreted in this way, Proposition 2 identifies a benefit of using a waterfall launch strategy (sequential release across markets), rather than a sprinkler strategy (simultaneous release in multiple markets), as the former strategy allows the firm to better exploit network effects.18 18
Note that Proposition 2 holds regardless of whether the firm sets the optimal price for each scheme or
simply sets the same price for both schemes.
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A sequential scheme provides consumers with increased information about each others’ purchases, essentially making their decisions visible to one another. This visibility can allow success to breed success. High sales from consumers who are served first can then encourage increased sales from consumers served later. The Proposition shows that sequential sales increase expected profits despite the fact that failure can also breed failure, where low early sales can depress sales from those who follow. The intuition for the result is as follows. With sequential sales, consumers not only observe earlier purchases, but they also realize their own purchases will be observed by later consumers. The very fact of being observed makes buying more attractive, since consumers who are served early understand that those who see them buy will become more likely to buy themselves. The key formal point is that the expectations in (5) for consumers in cohorts t0 ≥ t + 1 all condition on the purchase of consumer i in cohort t. It follows that a sequential strategy will tend to yield high initial sales, precisely because consumers are rational and forward looking, starting a virtuous cycle where early success is then compounded. An immediate corollary of Proposition 2 is that a fully simultaneous scheme is the worst possible choice. Every sales scheme is more sequential than a fully simultaneous scheme, according to Definition 1, and will give strictly higher expected profits. Our mechanism suggests that if a firm has any influence at all over sequencing, no matter how small, then it should use this influence to ensure that at least some consumers are served before others. Another consequence of Proposition 2 is that we can describe the optimal scheme. Corollary 1. The sales scheme I that maximizes expected profits has a single consumer per cohort. The optimal sales scheme is purely sequential with one consumer served after another. This result is relevant for a firm with perfect control over sequencing, for example one engaged in business-to-business transactions that can choose the precise order to approach potential clients. In this case, Corollary 1 says that the firm should approach clients one 16
by one. Unlike the literature on seeding, the firm does not attempt to kickstart product adoption by giving away its product to some clients to strengthen network effects. The firm instead tries to strengthen network effects and promote early sales by exploiting consumers’ strategic incentives to influence one another through their purchases.
3.2
Ordering of sales
The analysis in Section 3.1 shows that a firm can best exploit network effects by using a sales scheme that is as sequential as possible. But if not all feasible schemes are comparable in the sense of Definition 1, then the question remains as to precisely which scheme is best. For example, Proposition 2 suggests that a firm should launch its product sequentially across markets, but markets may well differ in their size. Should the firm then first release the product in smaller markets or in larger ones? In our setting, this amounts to asking whether the firm should use a scheme that serves smaller or larger cohorts first. Proposition 3. Suppose E(λ) < v0 − θ. Consider sales schemes I = {I1 , . . . , It , It+1 , . . . Im } 0 and I 0 = {I1 , . . . , It0 = It+1 , It+1 = It , . . . Im } with |It | ≡ nt > nt+1 ≡ |It+1 |. Then I 0 yields
strictly higher expected profits than I. This result provides additional insights into how to sequence sales when not all schemes are feasible. As discussed in Section 3.1, Proposition 2 suggests that a firm launching its product across multiple markets will benefit from selling sequentially across markets, so to one single market after another. Proposition 3 goes further by saying that the firm can best exploit network effects by carrying out this sequential launch in increasing order of market size. The intuition behind this result is that if larger markets move later, early consumer decisions influence a larger pool of consumers. Early movers understand that their actions influence the many consumers who follow and become more likely to buy themselves. Moving to our second result about ordering, we will relax the assumption that consumer type is private information. Intuitively, if a firm can observe certain consumer characteristics, 17
then it may well take this information into account when choosing its sales scheme. Corollary 1 showed that given complete control over sequencing, so if all schemes are feasible, then a firm should use a scheme is fully sequential. We now explore whether this remains the case when the seller can distinguish between different consumers, and in particular examine which consumers should be served first. The literature on word-of-mouth communication in networks has examined a related question from the point of view of consumer influence. Typically in this literature, a firm initially informs certain consumers about its product, these consumers pass this information along to others, those consumers pass this information along in turn, and so on. The issue for the firm is who to initially inform, in particular if it can distinguish between consumers with different propensities to pass along information. This propensity captures the strength of a consumer’s influence in the network.19 In our setting, all consumers are equally influential from an ex ante perspective, in the sense that the network payoff depend on total sales but not on the identity of those who buy. Not all consumers however are equally easy to influence. Consumers with high values of λ place a high weight on the network payoff which makes them more sensitive to others’ purchases. In contrast, consumers with low values of λ base their purchase decisions mainly on their intrinsic payoff from buying. To explore the issue of ordering and consumer influence, we now assume that each consumer’s value of λ is public information. The following result shows that the optimal sales scheme then serves consumers in increasing order of λ, so in decreasing order of their sensitivity to other consumers’ influence. Proposition 4. Suppose the weight consumers place on the network payoff, λ, is observable. Then the sales scheme I that maximizes expected profits has a single consumer per cohort, increasingly ordered in λ, i.e. λ1 ≤ . . . ≤ λn . 19
For example, Galeotti and Goyal (2009) suggest targeting influential consumers who will inform many
friends, whereas Campbell (2013) show this may be suboptimal if these influential consumers are likely to already be informed via word of mouth.
18
The optimal sales scheme remains purely sequential, as in Section 3.1, where the intuition for this result echoes that from Proposition 2 and Corollary 1. Just as the qualitative advantage of a sequential scheme did not depend on the precise distribution of λ, it does not depend on the exact realized values of these weights. As long as each consumer places a strictly positive weight on the network payoff, then a sequential scheme will increase expected profits by increasing visibility, pushing early consumers to buy, and allowing success to breed success. In addition, Proposition 4 derives the optimal ordering: consumers should be served in increasing order of the weight they place on the network payoff. This result complements those in the literature on word-of-mouth communication in networks stating that a firm should first serve consumers with the most influence. This result also echoes the notion that a firm launching a new product should target independent-minded consumers first, who can serve as opinion leaders for those who follow. These innovators (low λ) will decide whether or not to buy the product largely based on their own personal tastes. Their decision to buy can then encourage imitators (high λ) who care about their actions to jump on the bandwagon. When values of λ are observable, the seller faces a new trade off. Serving consumers in increasing order of these weights means that later consumers (high λ) have a strong incentive to follow those who buy. In principle, doing so reinforces the benefits when early consumers buy and success breeds success. However, these benefits are limited by the fact that early consumers (low λ) do not become much more likely to buy just because they expect others to follow. Another way to understand the trade off is that consumers with high weights are likely to set a good example, but they are also more likely to follow a good example once it has been set. Proposition 4 shows that the second effect outweighs the first so the optimal order is increasing in these weights. The mechanism behind sequential sales is based on the idea that consumers want to influence one another, but the optimal scheme grants the largest visibility to consumers who care the least about this influence.
19
Proposition 4 also relates to the broader economics literature on leadership, which has followed from Hermalin (1998). While precise definitions vary, this literature typically describes a leader as someone with potential followers, whose interests are at least partly aligned with his own and who can potentially be influenced by the leader’s actions. A leader typically wants to induce followers to coordinate both on his own and on each others’ actions. In our setting, given a purely sequential sales scheme, the leader corresponds to the consumer in the first cohort. This consumer can influence all those in later cohorts through his own purchase decision. Interests are partly aligned because of network effects, where the first consumer considers how his own decision to buy can help coordinate others to do so the same. Unlike Hermalin (1998), Dewan and Myatt (2008), and Bolton et al. (2013), who model leaders as having better information than followers, leadership in our setting come from a consumer’s position in the sales scheme. Moreover, the seller explicitly selects the sales scheme that will maximize consumers’ effective leadership, i.e. their ability to coordinate others on the seller’s preferred action (to buy the product). While the first consumer can be viewed as the leader, a sequential sales scheme also allows later consumers to show leadership, though to a lesser degree, by helping them influence those who come later still. Consumers are able to lead by example precisely because of the seller’s choice of a sequential scheme, which allows consumers to observe each others’ actions. By serving consumers in increasing order of λ, the seller select as leader the consumer who is least sensitive to others’ behavior. This result echoes that from both empirical work (Kaplan et al. 2012) and theoretical work (Goel and Thakor 2008, Bolton et al. 2013) showing that “resoluteness” can sometimes help with leadership. Resoluteness there is seen as a characteristic that makes one less susceptible to external influence, in the sense of often sticking with an early decision, regardless of others’ subsequent actions or of the arrival of additional information. Thus, in a broad sense, the leader in our setting, as selected by the seller, will be the consumer who is most resolute, in the sense of being least influenced by
20
the actions of others. Given this interpretation, Proposition 4 provides a novel link between resoluteness and leadership. Bolton et al. 2013 show that in the face of a time-inconsistency problem, a resolute leader may be able to commit to a particular action, which can facilitate follower coordination. In contrast, in our setting, the seller selects the most resolute consumer as leader because he wants consumers who are easily influenced to be followers. It is not that having a resolute leader helps in and of itself; to the contrary, the principal would prefer that all consumers were easily influenced by others (high λ), which would generate larger bandwagon effects. Rather, for a given group of consumers, what helps is that the leader should be more resolute, in a relative sense, than the followers. This is what maximizes the extent to which consumers coordinate on buying the good, to the benefit of the seller.
3.3
Consumer communication
Typically, models of sequential decision making with private information assume that consumers cannot directly communicate, and all information transmission takes place indirectly via observing each others’ purchases. For example, in the literature on quality uncertainty and social learning, consumers cannot directly share the private signal they receive about quality, and other consumers only update their beliefs about quality by observing the level of previous sales. We make a similar assumption in our analysis by assuming that consumers cannot directly communicate their willingness to pay. This assumption is reasonable in many situations where market interactions are anonymous. But for a variety of products, including books, films, mobile phones, and computers, consumers do share information in online forums and communities (see, e.g., Godes et al. (2005) and the references therein). This information sharing can pertain to new products that consumers have purchased, but also to products that are unreleased. For example, there are various online sites where consumers engage in heated debate about the perceived merits
21
of rumored Apple products that have yet to appear.20 Consumer communication is relevant in our setting because it may serve as a possible substitute for sequential sales. The whole purpose of sequential sales is to help consumers learn from one another about a product’s popularity. However, if consumers successfully learn each others’ willingness to pay through communication, then there is little scope for future learning, and little need for the visibility of purchases provided by sequential sales. Consumers who successfully communicate would be able to correctly predict the good’s popularity regardless of the seller’s choice of sales scheme. But a crucial point for communication to be successful relates to credibility. If consumers read certain comments or reviews about a product, should they actually believe what they read? One concern here is the potential for firm manipulation. Previous work has explored how a firm strategically post positive reviews about its own products to influence consumer beliefs; if consumers realize this, it will naturally reduce the credibility of the information they receive (Dellarocas (2006), Mayzlin et al. (2014)). In what follows we take an alternative approach focusing more on consumers. Rather than looking at firm manipulation, we examine another potential obstacle to credible communication: possible incentives for consumers to misreport. Specifically, we consider two reasons why consumers might want to misrepresent their willingness to pay to one another. On the one hand, information that consumers share may be collected by the firm and used to adjust the price (see, e.g., Chen and Xie 2008). A consumer who understates his willingness to pay may contribute to the impression that demand is low, leading to a price reduction. On the other hand, a consumer who overstates his willingness to pay may convince others to buy, and therefore himself benefit from a larger network payoff. This reasoning suggests that network effects might push consumers to overstate whereas firm monitoring might push them to understate. We now show that 20
See for example www.9to5mac.com and www.appleinsider.com
22
despite these potential obstacles, consumers may be able to communicate truthfully. Formally, we assume again that type is private information, but allow consumers to engage in cheap talk before making their purchase decisions. Consumers simultaneously send one another a message about their type, where the seller observes the set of messages with strictly positive probability. If the seller observes the messages then it can use this information when setting its price. The details of this price-setting process are not crucial for our results. The important point is just that the price be non-decreasing in the seller’s updated beliefs about consumer willingness to pay, conditional on observing the messages. Proposition 5. Consider a simultaneous sales scheme, with all consumers in the same cohort. Suppose that before buying, consumers can simultaneously send a message m ∈ [θ, θ] × (0, λ) about their type which all other consumers observe, and where the seller observes M = (m1 , . . . , mn ) with probability q > 0. Furthermore suppose that the seller sets price p∗ if it does not observe M , and sets price p(M ) if it does, where p(M ) is non-decreasing P in ni=1 E(xi |p∗ , M ). Then when q is sufficiently small, an equilibrium exists where communication is informative, in the sense that each consumer truthfully reveals to all others the minimum level of total sales required for him to buy himself at price p∗ . In the limit as q tends to zero, consumer purchase decisions approach those in a setting where consumers all observe each others’ type, (θi , λi ) for all i = 1, . . . , n. Proposition 5 shows that potential incentive problems need not rule out successful communication, in that consumers may still truthfully reveal their planned purchase behavior to one another. However, for such communication to occur, consumers must believe it sufficiently unlikely that the seller is monitoring their messages. Curiously enough, successful communication is possible precisely because of network effects, even though they seemingly provide consumers with an incentive to exaggerate. If there were no network effects, and the expected price was increasing in the stated willingness to pay in consumers’ messages, then consumers would all claim that their willingness to pay was very low, in the hopes of 23
obtaining a price reduction. As alluded to above, a consumer who understates his willingness to pay can generate two effects. The first effect is that other consumers infer demand may be relatively low, making them less likely to buy themselves, which reduces aggregate demand at any given price. The resulting reduction in the network payoff hurts consumers who buy. The second effect of understating is that the seller may reduce its price. This price reduction helps consumers who buy but can only occur if the seller observes the messages. When the probability of observing messages is relatively small, the first effect dominates the second, and consumers have a strict incentive not to understate. By a similar logic, overstating willingness to pay can push other consumers to buy, increasing the network payoff, but can also lead to a higher price. However, the consumer who overstates will only benefit from the increased network payoff if he has a genuine incentive to buy himself. And if he has such a genuine incentive, then there was no reason to overstate willingness to pay in the first place. The implication of Proposition 5 is that firms selling products where potential consumers regularly communicate may benefit less from a sequential product launch than firms for which the opposite is true. Successful communication can reduce uncertainty and leave consumers with relatively little to learn from one another through sequential sales. It may be tempting to also conclude that firms should actively facilitate discussion and encourage consumer communication about new products, for example through an official online forum or discussion board. However, if the firm’s involvement in this process leads consumers to suspect it is monitoring their messages, then this can derail successful communication, even absent any concern that the firm is strategically manipulating messages.
4
Discussion and Robustness
Our analysis has assumed that the seller commits to its choice of sales scheme, solutions are interior, and the intrinsic payoff θ is uniformly distributed. We now briefly comment on how 24
each of these assumptions relates to our results supporting sequential sales. We then address the issue of dynamic, rather than static, pricing. The fact that the seller can commit to a sales scheme is unimportant for the results. The analysis shows that for any cohort It with at least two consumers, given any history Kt , the seller always benefits by having some of these consumers act before the others. This means that a seller who chooses a sequential scheme at t = 0 has no incentive to change its mind after observing the actions of any number of consumers. If the first consumers do not buy, then the seller may well regret ex post using this scheme, but it will still prefer the remaining consumers to act sequentially. Our assumption on parameter values ensures that after any history, the probability a consumer will buy lies strictly between zero and one, so that the equilibrium of the consumer game is interior and unique. Relaxing this assumption would mean that multiple values of expected profits could be consistent with each scheme, as discussed following Proposition 1. If network effects were sufficiently strong, then any sales scheme could generate both a good equilibrium outcome where all consumers buy and a bad equilibrium outcome where nobody buys. It would then be difficult to rank different schemes, but sequential sales might still be useful in helping with equilibrium selection, if observing an early purchase can coordinate the remaining consumers on the Pareto dominant outcome. Assuming a uniform distribution of θ guarantees equilibrium existence and uniqueness, as discussed prior to Proposition 1. It also has an effect that relates to the variance of early sales. Intuitively, variance can be quite high under a sequential scheme, since consumers can observe and imitate one another. This reasoning suggests that in comparison with simultaneous sales, a sequential scheme may tend to generate more extreme histories. The variance of early sales plays no role when θ is uniformly distributed. All that matters about early sales is their expected value, which is maximized under a sequential scheme. However, variance could potentially matter if θ followed a different distribution. For example,
25
if many consumers had low θ, and only a very good history would persuade them to buy, then high variance could help by increasing the probability of such a history. If instead many consumers had high θ, so only a very bad history would dissuade them from buying, then high variance could hurt by the same reasoning. Our analysis would then underestimate the benefit of sequential sales in the first case but overestimate it in the second case. We now turn to dynamic pricing and address whether sequential sales will remain attractive to a seller that can adjust its price over time. For example, the seller may increase the price from its initial level if early sales are high or decrease the price if early sales are low. A fully general analysis of dynamic pricing is complicated by the fact that the seller jointly chooses a price schedule and a sales scheme, and the preferred prices will vary across different schemes. For any given sales scheme, the analysis would involve considering all potential prices to charge each cohort, for every possible history, and then comparing the resulting profits across all possible schemes. An additional complication is that the optimal schedule will depend on whether the seller can commit to future prices. Commitment means that the seller fixes a price schedule at t = 0 so as to maximize expected profits from an ex ante perspective. No commitment means that the seller effectively makes a sequence of pricing decisions over time when facing each cohort, where the chosen price must maximize expected profits from that particular cohort and all later consumers, given observed sales. We analyze the former case in the following Proposition and leave the latter for further research. Proposition 6. Suppose the seller can commit to a dynamic pricing schedule, with price p(Kt ) for cohort t conditional on previous sales Kt . Then a fully sequential sales scheme (a single consumer per cohort), delivers higher expected profits than a fully simultaneous sales scheme (all consumers in a single cohort). Fully sequential sales remain more profitable than fully simultaneous sales under dynamic pricing, just as under static pricing. Dynamic pricing actually increases the difference in expected profits between these schemes because sequential sales now offer additional flexibility, 26
allowing the seller to adjust the price depending on whether early sales were high or low. With dynamic pricing, the seller can always earn the same profits as under static pricing by maintaining its initial price, but can generally do better still by adjusting its price over time. We conclude this section with a numerical example. Section 3.1. show that the ranking of sales schemes is independent of the exact distribution of λ, the weight consumers place on the network payoff. However, the distribution of λ does have a quantitative impact on the strength of the link between sequential sales and expected profits. Sequential sales help the seller by exploiting network effects between consumers. The stronger these effects, the larger the impact we would intuitively expect from a sequential scheme. Figure 1 below provides support for this idea. For all i, we set λi = λ with probability one, where the figure presents expected profits under both a fully simultaneous and a fully sequential scheme as a function of λ. Figure 1 Expected Profits as a Function of Network Effect Strength 1.2
ExpectedProfits
1.0
0.8
0.6
0.4 0.0
0.2
0.4
0.6
0.8
1.0
l
Expected profits with simultaneous sales are represented in blue and those with sequential sales are given in in red. As required by Corollary 1, expected profits are higher under a 27
sequential scheme for all values of λ > 0. The difference in expected profits between the two schemes is increasing in λ, consistent with intuition provided above.21 Despite the notion that sequential sales may expose the seller to downside risk, where early failure breeds further failure, we now show that a sequential scheme may nonetheless dominate a simultaneous scheme, in the sense of first order stochastic dominance. Figure 2 CDF of Profits for Simultaneous and Sequential Schemes
CumulativeProbability
1.0
0.8
0.6
0.4
0.2
0
1
2
3
4
5
Profits
Figure 2 plots the cumulative distribution function of total profits under the two different schemes. The simultaneous scheme is represented in blue and the sequential scheme in red, where the former CDF lies entirely above the latter.22 A sequential scheme here serves the dual purpose of increasing expected profits while decreasing the probability of a poor outcome where realized profits are very low. It is true that low early early sales under a sequential scheme can dissuade later consumers from buying. However, the positive incentive 21 22
The simulation uses parameter values n = 5, θ = 2, θ = 0, u0 = 1.85, and p = 1 under both schemes. The simulation uses parameter values n = 5, θ = 2, θ = 0, u0 = 1.85, p = 1, and λ = 1 under both
schemes.
28
effect of a sequential scheme on early consumers is so strong that it outweighs any increased risk that might arise from low early sales.
5
Conclusion
In a setting with network effects, consumers looking to buy a product will naturally take into account the expected purchase behavior of others. They may be more willing to buy an electric car or a mobile phones, movies tickets or books, if they believe that others will buy as well. For this reason, a firm selling a product that exhibits network effects would like consumers to believe it will likely become a ‘hit’. We examine how the sequencing of sales can affect such beliefs, and allow consumers to lead by example, to best exploit network effects for the firm’s own benefit. We show that in our setting, a firm can always increase its expected profits by moving from one sales scheme to another that is more sequential. The key point is that consumers are rational and forward looking, and a sequential scheme affects their expectations about how others will behave in the future. A consumer knows that others who observe his purchase will become more likely to buy, which increases the incentive to buy himself. The use of a sequential scheme not only reveals to consumers whether or not the product is a hit, it also makes a hit more likely in the first place. Our results identify an advantage of using a sequential product-launch strategy, where a firm first releases its product in smaller markets before moving on to larger ones. They also suggest that a firm with full control of sequencing should approach potential clients in a way that is purely sequential, one after the other. Such a firm will also benefit from first serving more independent-minded consumers, who are less sensitive to other consumers’ behavior, and who can serve as opinion leaders for those who follow.
29
6
Appendix*
6.1
Technical Lemmas*
Lemma A.1. Suppose Kt consumers buy up until cohort It , and consider consumer j ∈ It0 0
−1 with t0 ≥ t + 1. Suppose a set of consumers M ⊆ ∪tl=t Il choose to buy. Then
θ − E(θj∗ |Kt , M )
E(xj |Kt , M ) =
θ−θ
,
where E(θj∗ |Kt , M ) = u0 −
E(λ) Kt + n−1 t≤l≤t0 X
X
E(xi |Kt , M ) +
X X l≥t0 +1
i∈Il \{j}
E(xi |Kt , M, xj = 1) .
i∈Il
Proof. By (5), for any Kt0 , the relevant cutoff for consumer j ∈ It0 is X X X E(λ) E(xi |Kt0 ) + E(xi |Kt0 , xj = 1) , Kt0 + θj∗ = u0 − n−1 0 l≥t +1 i∈I i∈It0 \{j}
(6)
l
so that E(xj |Kt0 ) =
θ − θj∗ θ−θ
.
(7)
We now work with (6) and (7) to obtain E(xj |Kt , M ). Let K be the set of all Kt0 consistent with (Kt , M ). For each Kt0 we multiply (6) with p(Kt0 |Kt , M ) and sum up over all Kt0 ∈ K. P Since K p(Kt0 |Kt , M ) = 1, we have that E(θj∗ |Kt , M ) is equal to u0 −
X X E(λ) X Kt0 p(Kt0 |Kt , M ) + p(Kt0 |Kt , M )E(xi |Kt0 )+ n−1 K K i∈It0 \{j} ! X XX p(Kt0 |Kt , M )E(xi |Kt0 , xj = 1) , l≥t0 +1 i∈Il
K
Note that
E(xj |Kt , M ) =
X
p(Kt0 |Kt , M )E(xj |Kt0 ),
K
30
E(xi |Kt , M, xj = 1) =
X
p(Kt0 |Kt , M )E(xj |Kt0 , xj = 1),
K
and
P
K
Kt0 p(Kt0 |Kt , M ) = Kt +
P
t≤l≤t0 −1
P
i∈Il
E(xi |Kt , M ). Therefore,
E(θj∗ |Kt , M ) = u0 −
X X X X E(λ) E(xi |Kt , M, xj = 1) . Kt + E(xi |Kt , M ) + n−1 0 0 t≤l≤t l≥t +1 i∈I i∈Il \{j}
l
Lemma A.2. For any consumer i in cohort It with history Kt , dE(xi |Kt ) > 0. dKt Proof. Write out E(xi |Kt ) =
θ−θi∗ θ−θ
with
θi∗ = u0 −
X X X E(λ) E(xj |Kt , xi = 1) . Kt + E(xj |Kt ) + n−1 0 t ≥t+1 j∈I j∈It \{i}
t0
By Lemma A.1, write out each term in the second summation as E(xj |Kt , xi = 1) = θ−E(θj∗ |Kt ,xi =1) θ−θ
with E(θj∗ |Kt , xi
X E(λ) = 1) = u0 − Kt + 1 + E(xj 0 |Kt ) n−1 0 j ∈It \{i}
+
X
X
E(xj 0 |Kt , xi = 1) +
t+1≤l≤t0 j 0 ∈Il \{j}
X X
E(xj 0 |Kt , xi = 1, xj = 1) .
l≥t0 +1 j 0 ∈Il
Again by Lemma A.1, write out each term E(xj 0 |Kt , xi = 1, xj = 1) in the last summation as E(xj 0 |Kt , xi = 1, xj = 1) =
θ−E(θj∗0 |Kt ,xi =1,xj =1) θ−θ
, and so on. Consider a player in a cohort
k > t + 1. Let Mk be the subset of players such that (i) each player i ∈ Mk decided to buy, (ii) for i, j ∈ Mk , i ∈ Ini , j ∈ Inj ni 6= nj and ni > t. Let for l ≤ k Mkl ⊆ Mk : ∀i ∈ Mkl , i ∈ Ini ⇒ ni < l. Then E(θj∗ |Kt , xi
X E(λ) = 1, Mk ) = u0 − Kt + 1 + #Mk + E(xj 0 |Kt ) n−1 0 j ∈It \{i}
31
+
X t+1≤l≤k
X j 0 ∈I
E(xj 0 |Kt , xi =
1, Mkl )
+
X X
E(xj 0 |Kt , xi = 1, Mk , xj = 1) .
l≥k+1 j 0 ∈Il
l \Mk
Denoting the number of distinct equations for E(xj |Kt , xi = 1, Mk ) by A, including terms with zero coefficient on the right-hand side of each equation, gives a system of A equations in A unknowns. As shown immediately after (4) in Section 3, consumers with θ sufficiently close to θ have a dominant strategy to buy. This means any solution to this system must give E(xi |Kt ) > 0, for any Kt .
dθ ∗
Differentiating each equation in the system with respect to Kt gives
dE(xi |Kt ) dKt
=
θ− dKi
t
θ−θ
with X dE(θj∗ |Kt , xi = 1, Mk ) E(λ) = u0 − 1+ dKt n−1 0
j ∈It \{i}
+
X
X
t+1≤l≤k j 0 ∈Il \Mk
dE(xj 0 |Kt ) dKt
X X dE(xj 0 |Kt , xi = 1, Mk , xj = 1) dE(xj 0 |Kt , xi = 1, Mkl ) + . dKt dKt l≥k+1 j 0 ∈I l
This system of A linear equations in A unknowns is identical to the first one, except that each conditional expectation is replaced by its derivative, and Kt has been set equal to 1. The associated matrix for this system has diagonal entries of 1 and off-diagonal entries of −1 E(λ) < 0, where the number of non-zero off-diagonal entries in each row cannot either 0 or θ−θ n−1 P P exceed t0 ≥t i∈It0 ni − 1 ≤ n − 1. By E(λ) ≤ λ = θ − vo and θ + 1 < vo < θ, the sum of the
absolute values of off-diagonal entries in each row is therefore strictly less than one. Hence, this matrix is strictly diagonally dominant. By the Gershgorin theorem (1931), the system then has a unique solution, with
dE(xi |Kt ) dKt
> 0.
Lemma A.3. For any consumer j ∈ It0 , with t0 ≥ t + 1, E(xj |Kt ) is strictly increasing in P i∈It E(xi |Kt ). Proof. We proceed by induction. First, let t0 = t + 1. By Lemma 1, for any xj ∈ It+1 , write
32
out E(xj = 1|Kt ) =
θ−E(θj∗ |Kt ) θ−θ
with
"
E(θj∗ |Kt ) = u0 −
#
X E(λ) Kt + E(xi |Kt ) + n−1 i∈I t
X
E(xi |Kt ) +
X X
E(xi |Kt , xj = 1) .
l≥t+2 i∈Il
i∈It+1 \{j}
Again by Lemma A.1, write out each expectation E(xi |Kt , xj = 1) in the last summation, and so on to generate a system of equations. Each of these equations will include the same expression in square brackets. We can identify the expression in square brackets with a constant Kt+1 . A strict increase P in i∈It E(xi |Kt )) is then equivalent to a strict increase in Kt+1 . Hence by Lemma A.2, E(xj |Kt ) must strictly increase. Now let t0 ≥ t + 2, and suppose the result holds for all cohorts t + 1, . . . , t0 − 1. We show that the result also holds for t0 . For a consumer j ∈ It0 , write # " X X E(λ) E(xi |Kt ) + Kt + E(θj∗ |Kt ) = u0 − n−1 0 t≤l≤t −1 i∈I l
X
E(xi |Kt ) +
X X l≥t0 +1
i∈Il \{j}
E(xi |Kt , xj = 1) .
i∈Il
Once again using Lemma A.1, write out each expectation E(xi |Kt , xj = 1) in the last summation, and so on to generate a system of equations which all include the same term in square brackets. By the induction hypothesis, the term in square brackets strictly increases, which is again equivalent to an increase in Kt0 . By Lemma A.2, E(xj |Kt ) must strictly increase.
Lemma A.4. Let t0 ≤ t − 1. Consider a history Kt0 and any consumer i ∈ Il with l ≥ t0 . Let Kt0 be a set of histories Kt consistent with Kt0 , and let a ∈ R be some parameter of arbitrary nature. Then if
d
P
E(xj |Kt ) da
j∈It
> 0 for all Kt ∈ Kt0 , then
E(xi |Kt0 ) da
> 0.
Proof. First let t0 = t − 1, and consider a consumer i ∈ It−1 , given history Kt−1 . By (4) and
33
(5), write E(xi |Kt−1 ) = θi∗ = u0 −
E(λ) n−1
θ−θi∗ θ−θ
with
Kt−1 +
X j∈It−1 \{i}
X
E(xj |Kt−1 ) +
P(
0≤K 0 ≤#It−1 −1
X
xj = K 0 |Kt−1 )
XX
E(xj |Kt−1 + 1 + K 0 ) ,
l≥t j∈Il
j∈It−1 \{i}
explicitly writing out all the possible histories Kt consistent with Kt−1 and xi = 1. Each such P history corresponds to a value of j∈It−1 \{i} xj = K 0 , with K 0 = 0, . . . , #It−1 −1, representing the possible purchase decisions of the #It−1 − 1 consumers in It−1 \ {i}. Equivalently, consumer i will find it optimal to buy if and only if his expected payoff from buying, X λi Kt−1 + E(xj |Kt−1 ) + θi + n−1 j∈It−1 \{i} X X XX P( xj = K 0 |Kt−1 ) E(xj |Kt−1 + 1 + K 0 ) , 0≤K 0 ≤#It−1 −1
(8)
l≥t j∈Il
j∈It−1 \{i}
exceeds that from his effective outside option, u0 . P Consider an increase in j∈It E(xj |Kt ) for every history Kt consistent with Kt−1 . This P implies an increase in j∈It E(xj |Kt−1 +1+K 0 ) for all K 0 = 0, . . . #It−1 −1. Then by Lemma P P A.3, l≥t+1 j∈Il E(xj |Kt−1 +1+K 0 ) must increase as well, for all such K 0 . Since λ > 0 for all consumers, the system of equations given by (8) defines a game with strategic complements between all consumers i in cohort It−1 (increasing best-response functions). Therefore, if P P θ−θi∗ 0 l≥t j∈Il E(xj |Kt−1 + 1 + K ) increases, then for each i ∈ It−1 , E(xi |Kt−1 ) = θ−θ must P increase as well (see Vives (1990)). Given this increase in i∈It−1 E(xi |Kt−1 ), Lemma A.3 implies that E(xj |Kt−1 ) must also increase, for any consumer j in cohort l ≥ t. Proceeding by induction for cohorts t0 = t − 2, t − 3, . . . , 1 completes the proof.
34
6.2
Proofs of the Propositions*
Proposition 1. For any sales scheme I, the game has a unique perfect bayesian equilibrium. That is, for any consumer i ∈ It and history Kt , the cut-off function θi∗ (λi ) is uniquely defined. Proof. Consider a subgame starting with cohort It to act after a history summarized by Kt . By (4) and (5), for each consumer i ∈ It , write out E(xi |Kt ) =
θ−θi∗ , θ−θ
with
θi∗ = u0 −
E(λ) Kt + n−1
X
E(xj |Kt ) +
X X
E(xj |Kt , xi = 1) .
t0 ≥t+1 j∈It0
j∈It \{i}
By Lemma A.1, write out each term in the second summation as E(xj |Kt , xi = 1) = θ−E(θj∗ |Kt ,xi =1) θ−θ
, with E(θj∗ |Kt , xi
X E(λ) = 1) = u0 − Kt + 1 + E(xj 0 |Kt ) n−1 0 j ∈It \{i}
X
+
X
E(xj 0 |Kt , xi = 1) +
t+1≤l≤t0 j 0 ∈Il \{j}
X X
E(xj 0 |Kt , xi = 1, xj = 1) .
l≥t0 +1 j 0 ∈Il
Again by Lemma A.1, write out each term E(xj 0 |Kt , xi = 1, xj = 1) in the last summation as E(xj 0 |Kt , xi = 1, xj = 1) =
θ−E(θj∗0 |Kt ,xi =1,xj =1) θ−θ
, and so on. Consider a player in a cohort
k > t + 1. Let Mk be the subset so players such that (i) each player i ∈ Mk decided to buy, (ii) for i, j ∈ Mk , i ∈ Ini , j ∈ Inj ni 6= nj and ni > t. Let for l ≤ k Mkl ⊆ Mk : ∀i ∈ Mkl , i ∈ Ini ⇒ ni < l. Then E(θj∗ |Kt , xi
X E(λ) = 1, Mk ) = u0 − Kt + 1 + #Mk + E(xj 0 |Kt ) n−1 0 j ∈It \{i}
+
X
X
t+1≤l≤k j 0 ∈Il \Mk
E(xj 0 |Kt , xi =
1, Mkl )
+
X X
E(xj 0 |Kt , xi = 1, Mk , xj = 1) .
l≥k+1 j 0 ∈Il
Denoting the number of distinct equations for E(xj |Kt , xi = 1, Mk ) by A, including terms with zero coefficient on the right-hand side of each equation, gives a system of A equations in A unknowns. 35
The associated matrix for this system has diagonal entries of 1 and off-diagonal entries of either 0 or
−1 E(λ) θ−θ n−1
< 0, where the number of non-zero off-diagonal entries in each row P P cannot exceed t0 ≥t i∈It0 ni − 1 ≤ n − 1. By E(λ) ≤ λ = θ − vo and θ + 1 < vo < θ, the sum of the absolute values of off-diagonal entries in each row is therefore strictly less than one. Hence, this matrix is strictly diagonally dominant. By the Gershgorin theorem (1931), the system then has a unique solution. In particular, this unique solution implies that E(xj |Kt ) for each consumer j 6= i in cohort t, and E(xj |Kt , xi = 1) for each consumer j in cohort t0 ≥ t + 1, are all uniquely defined. Hence, the cut-off function for consumer i,
θi∗ (λ) = u0 −
λi Kt + n−1
X
E(xj |Kt ) +
X X
E(xj |Kt , xi = 1) ,
t0 ≥t+1 j∈It0
j∈It \{i}
given by (3) is uniquely defined as well.
Proposition 2. Suppose that a sales scheme I’ is more sequential than another scheme I, according to Definition 1. Then I’ delivers strictly higher expected profits. Proof. We prove the following result, where repeated application given Definition 1 will immediately imply Proposition 2: Consider sales schemes I = {I1 , . . . , It−1 , It , It+1 , . . . Im } and I’ = {I1 , . . . , It−1 , It0 , It00 , It+1 , . . . Im }, where It = It0 ∪ It00 . Then I’ delivers strictly higher expected profits. Let p denote the optimal price under I. Suppose for now that the seller charges p under both schemes, so that both I and I’ involve the same net outside option, u0 ≡ v0 + p. Suppose that under I’, there are l consumers in cohort It0 . Denote these consumers by subscript i, for i = 1, . . . , l. Under I, these consumers are all members of cohort It ⊇ It0 , and the probability that they will buy, given history Kt , is characterised by cut-off X X X E(λ) θi∗ = u0 − Kt + E(xj |Kt , θ) + E(xj |Kt , xi = 1, θ) , n−1 t0 ≥t+1 j∈I j∈It \{i}
t0
36
(9)
where θ = {θ1∗ , . . . , θl∗ } is the vector of cutoffs for these l consumers; E(·|Kt , θ) is the expectation conditional on history Kt and the fact that these l consumers have cut-offs θ. Due to Proposition 1, there exists a unique vector θ resulting from consumer optimizing behavior, given Kt and I. In fact, (9) implies θ1∗ = . . . = θl∗ , but our notation allows for the fact that cutoffs will differ if λ is observable, in which case λi will replace E(λ) in (9). Now, under I’, It is split into two cohorts, It0 and It00 . For the l consumers in cohort It0 , the probability that they will buy, given history Kt , is characterised by cut-off X X E(λ) 0 Kt + E(xj |Kt , θ 0 ) + E(xj |Kt , xi = 1, θ 0 )+ θi∗ = u0 − n−1 j∈It00 j∈It0 \{i} X X E(xj |Kt , xi = 1, θ 0 ) ,
(10)
t0 ≥t+1 j∈It0 0
0
where θ 0 = {θ1∗ , . . . , θl∗ } is the vector of cutoffs for these l consumers; E(·|Kt , θ 0 ) is the expectation conditional on history Kt and the fact that these l consumers have cut-offs θ 0 . Again due to Proposition 1, there exists a unique vector θ resulting from consumer optimizing behavior, given Kt and I’. We now use the Jacobi iterative method to show that θ 0 < θ; that is to say θi∗ 0 < θi∗ for i = 1, . . . , l. This method consists of plugging an initial approximation for θ 0 into the system of equations determining the cutoffs under I’, solving for the cutoffs θ 0 1 that are then implied by these equations, where θ 0 1 may well differ from θ 0 0 , and repeating the process with θ 0 1 , θ 0 2 , . . . . Recall from the proof of Proposition 1 that the system of equations determining the cutoffs is strictly diagonally dominant, which implies that given any initial approximation θ 0 0 , the iterations must converge to the unique fixed point θ 0 of the system (see, e.g., Varga (1962)). Hence, to show θ 0 < θ, it is sufficient to find θ 0 0 such that θ 0 n < θ holds for every iteration n ≥ 1, and to show that the iterative process does not converge to exactly θ.
37
0
t ,xi =1,θ ) For each consumer k in cohort It00 write E(xk |Kt , xi = 1, θ 0 ) = θ−E(θk |K , where θ−θ X E(λ) E(θk |Kt , xi = 1, θ 0 ) = u0 − [Kt + 1] + E(xj |Kt , θ 0 )+ n−1 j∈It0 \{i} (11) X X X E(xj |Kt , θ 0 , xi = 1) + E(xj |Kt , xi = 1, θ 0 , xi = xk = 1) .
j∈It00 \{k}
t0 ≥t+1 j∈It0
Let θ 0 0 = θ. By assumption, the behavior of consumers i ∈ It0 is then the same as under I. From (11), the decision problem of consumers k ∈ It00 is the same as under I, but with E(xi |Kt , θ) < 1 replaced by 1. Lemma A.2 and Lemma A.3 then imply E(xj |Kt , xi = 1, θ 0 0 ) > E(xj |Kt , θ) for all consumers in cohorts It00 , . . . , Im . From (10), this in turn implies θ 0 1 ≡ R(θ 0 0 ) < θ 0 0 , hence θ 0 1 < θ. It follows that the iterative process cannot converge to exactly θ, since that would require θ 0 1 = θ. Now assume θ 0 n < θ for some iteration n ≥ 1, so that X
E(xj |Kt , θ 0 n ) >
j∈It0 \{i}
X
E(xj |Kt , θ).
j∈It0 \{i}
From (11), the decision problem of consumers k ∈ It00 is the same as under I, but with P P E(xi |Kt , θ) < 1 replaced by 1, and with j∈It0 \{i} E(xj |Kt , θ) replaced by j∈It0 \{i} E(xj |Kt , θ 0 n ). Lemma A.2 and Lemma A.3 then imply E(xj |Kt , xi = 1, θ 0 n ) > E(xj |Kt , θ) for all consumers in cohorts It00 , . . . , Im . From (10), this in turn implies θ 0 n+1 ≡ R(θ 0 n ) < θ. It follows by induction that θ 0 n < θ holds for every iteration n ≥ 1, as required. The results so far show that conditional on any given history Kt , moving to I’ will P cause j∈It0 E(xj |Kt ) to increase. For each consumer k in cohort It00 write E(xk |Kt , θ 0 ) = θ−E(θk |Kt ,θ 0 ) , θ−θ
where E(θk |Kt , θ 0 ) =
u0 −
E(λ) [Kt + n−1
X j∈It0
E(xj |Kt , θ 0 )] +
X
E(xj |Kt , θ 0 ) +
j∈It00 \{k}
X X
E(xj |Kt , θ 0 , xk = 1) .
t0 ≥t+1 j∈It0
(12) P Looking at the term in square brackets, moving to I’ is equivalent to replacing j∈It0 E(xj |Kt , θ) P 0 by the strictly larger j∈It0 E(xj |Kt , θ ). This is in turn equivalent to replacing history 38
P E(xj |Kt , θ) by the strictly larger Kt + j∈It0 E(xj |Kt , θ 0 ), so it follows from P Lemma A.2 and Lemma A.3 that j∈It00 E(xj |Kt ) will increase.
Kt +
P
j∈It0
Hence, for any history Kt , moving to I’ increases expected total sales from consumers preP P P viously in cohort t, from j∈It E(xj |Kt ) to j∈It0 E(xj |Kt ) + j∈It00 E(xj |Kt ). Thus Lemma A.4 with t0 = 1 implies that E(xj ) strictly increases for all consumers. Hence, ex ante exP pected profits, p nj=1 E(xj ) are strictly higher under I’ than under I, given the assumption that the seller charges price p under both schemes. Let p0 denote the optimal price under I’. By the optimality of this price, ex ante expected profits under I’ at price p0 must be strictly higher than expected profits under I at price p.
Proposition 3. Suppose E(λ) < v0 − θ. Consider sales schemes I = {I1 , . . . , It , It+1 , . . . Im } 0 = It , . . . Im } with |It | ≡ nt > nt+1 ≡ |It+1 |. Then I 0 yields and I 0 = {I1 , . . . , It0 = It+1 , It+1
strictly higher expected profits than I. Proof. The proof is similar to that of Proposition 2. We first fix some history Kt and the expected actions of consumers in cohorts It+2 and after. We then show that swapping It and It+1 will strictly increase expected sales from these two cohorts, conditional on this history. Finally, direct application of Lemmas A.4 with t0 = 1 implies that from an ex-ante perspective, expected sales from all cohorts will strictly increase. Let p denote the optimal price under I. Suppose for now that the seller charges p under both schemes, so that both I and I 0 involve the same net outside option, u0 ≡ v0 + p. P P Denote L = Kt + 1 + l≥t+2 j∈Il E(xj |Kt ). From the perspective of consumer i in cohort t, following history Kt , L is the expected number of total sales, ignoring the behavior of consumers in cohorts It and It+t . Its value will depend on the expected behavior of others in cohorts It and It+1 , but for now this dependence is left implicit. Then, holding L constant, Lemma A.1 implies that the expected cut-off θt+1 for consumers in cohort It+1 ,
39
from the perspective of consumer i in It who buys, is determined by: E(λ) θt+1 + L + (nt − 1)E(xj |Kt ) + (nt+1 − 1)E(xj |Kt , xi = 1) = u0 , n−1 which can be rewritten as E(λ) θt+1 + n−1
θ − θt θ − θt+1 L + (nt − 1) + (nt+1 − 1) = u0 . θ−θ θ−θ
(13)
Expression (13) shows that θt+1 depends on θt , which is the expected cut-off for a consumer in cohort It , conditional on Kt . This cutoff θt is determined in turn by θ − θt θ − θt+1 E(λ) L − 1 + (nt − 1) + nt+1 = u0 . θt + n−1 θ−θ θ−θ
(14)
From the perspective of the seller, expected sales from cohorts It and It , conditional on history Kt , are then S(nt , nt+1 ) = nt
0 θ − θt+1 θ − θt + nt+1 , θ−θ θ−θ
0 is defined by where Lemma A.1 implies that θt+1
0 θt+1 +
E(λ) n−1
0 θ − θt+1 θ − θt L − 1 + nt + (nt+1 − 1) θ−θ θ−θ
! = u0 .
(15)
0 is the expected cutoff for consumers in cohort It+1 , from the perspective of the That is, θt+1
seller serving cohort It , conditional on history Kt . Solving the system (13)–(15) allows us to determine the cut-offs and compute S(nt , nt+1 ). Let ∆ = S(nt , nt+1 ) − S(nt+1 , nt ). Then, ∆=
(nt+1 − nt )(θ − θ)[(n − 1)(θ − θ) + E(λ)][(n − 1)(u0 − θ) − (L + nt + nt+1 − 2)E(λ)] . 1 G · G2 · G3 · G4 nt nt+1 (n−1)(E(λ))3 1
where G1 = (n − 1)(θ − θ) − (nt+1 − 1)E(λ), G2 = (n − 1)(θ − θ) − (nt − 1)E(λ), G3 = (n − 1)2 (θ − θ)2 − (n − 1)(nt + nt+1 − 2)(θ − θ)E(λ) − (nt+1 − 1)(E(λ))2 , G4 = (n − 1)2 (θ − θ)2 − (n − 1)(nt + nt+1 − 2)(θ − θ)E(λ) − (nt − 1)(E(λ))2 . 40
Due to E(λ) ≤ θ − v0 < θ − θ, G3 can be rewritten as G3 = (n − 1)(θ − θ) (n − 1)(θ − θ) − (nt + nt+1 − 2)E(λ) − (nt+1 − 1)(E(λ))2 ≥ (θ − θ)2 [(n − 1)(n − 1 − nt − nt+1 + 2) − nt+1 + 1] ≥ (θ − θ)2 (n − 1 − nt + 1) > 0. In a similar fashion G1 , G2 and G4 are all positive. Note that as u0 − θ > v0 − θ > E(λ) holds by assumption, and L − 1 + nt + nt+1 ≤ n, the last term in the numerator of ∆ is always positive, which implies that ∆ > 0 whenever nt+1 > nt . Thus, compared with I, sales scheme I 0 yields strictly higher expected sales from cohorts It and It+1 , conditional on Kt . Thus Lemma A.4 with t0 = 1 implies that the ex ante probability of buying, E(xj ), strictly P increases for all consumers. Hence, ex ante expected profits, p nj=1 E(xj ), are strictly higher under I’ than under I, if the seller charges price p under both schemes. Let p0 denote the optimal price under I’. By the optimality of this price, ex ante expected profits under I’ at price p0 must be strictly higher than expected profits under I at price p.
Proposition 4. Suppose the weight consumers place on the network payoff, λ, is observable. Then the sales scheme I that maximizes expected profits has a single consumer per cohort, increasingly ordered in λ, i.e. λ1 ≤ . . . ≤ λn . Proof. Notice first that all previous results continue to hold when λ is observable. Proofs remain unchanged except that the relevant cutoff for any consumer i is now θi∗ (λi ) given by (3), rather than θi∗ ≡ Eλ (θi∗ (λi )) given by (5). That is, the only difference is that λi replaces E(λ) in the expression for this cutoff. Hence, by Corollary 1, the sales scheme that maximizes expected profits still has a single consumer per cohort. For the optimal ordering of these consumers, we prove the result directly. Consider a fully sequential partition with one consumer per cohort, and fix p at the optimal price for this partition. Consider two subsequent consumers: i and i + 1. Suppose there where K
41
consumers who bought before consumer i. Then P i+1 θ − u0 + λn−1 K + 1 + nj=i+2 E(xj |K, xi = xi+1 = 1) E(xi+1 |K, xi = 1) = , θ−θ θ − u0 + E(xi+1 |K, xi = 0) =
λi+1 n−1
K+
E(x |K, x = 0, x = 1) j i i+1 j=i+2
Pn
θ−θ
.
Now we look at consumer i, where E(xi |K) =
1 θ−θ
θ − u0 +
λi n−1
K+
P(xi+1 = 1|K, xi = 1) 1 +
n X
! E(xj |K, xi = xi+1 = 1) +
j=i+2
P(xi+1 = 0|K, xi = 1)
n X
!! E(xj |K, xi = 1, xi+1 = 0)
.
j=i+2
Clearly in our setting P(xi+1 = 1|K) = E(xi+1 |K), for any history K. Now define: S(λi , λi+1 ) ≡
n X
E(xj |K) =
j=i
P(xi = 1|K) P(xi+1 = 1|K, xi = 1) 2 +
n X
! E(xj |K, xi = xi+1 = 1) +
j=i+2
P(xi+1 = 0|K, xi = 1) 1 +
n X
!! E(xj |K, xi = 1, xi+1 = 0)
+
j=i+2
P(xi = 0|K) P(xi+1 = 1|K, xi = 0) 1 +
n X
! E(xj |K, xi = 0, xi+1 = 1) +
j=i+2
P(xi+1 = 0|K, xi = 0)
n X
! E(xj |K, xi = 0, xi+1 = 0) .
j=i+2
We now use the fact that all expectations are linear in prior sales: E(xj |K, xi = 0, xi+1 = 1) = E(xj |K, xi = 1, xi+1 = 0) and 2E(xj |K, xi = 1, xi+1 = 0) = E(xj |K, xi = 0, xi+1 = 0) + E(xj |K, xi = xi+1 = 1). Thus S(λi , λi+1 ) − S(λi+1 , λi ) = −
(2 + Q2 − Q0 )3 (Q2 + K + 1)(λi − λi+1 )λi λi+1 , 8(n − 1)3 (θ − θ)3 42
(16)
where Q2 ≡
n X
E(xj |K, xi = xi+1 = 1),
j=i+2
Q0 ≡
n X
E(xj |K, xi = xi+1 = 0).
j=i+2
Lemma A.2 implies that Q2 > Q0 . Hence by (16), we have S(λi , λi+1 ) − S(λi+1 , λi ) > 0 if and only if λi < λi+1 . If λi > λi+1 , then allowing consumer i + 1 to act before consumer i will P strictly increase nj=i E(xj |K), for any history K. Hence, applying Lemma A.4 with t0 = 1, P allowing consumer i + 1 to act before consumer i will also strictly increase 1≤i≤n E(xi ). P It follows that ex ante expected profits, p 1≤i≤n E(xi ), are strictly higher under this new ordering, where p was the optimal price under the original ordering. Let p0 denote the optimal price under the new ordering. Thus, by optimality of this price, ex ante expected profits under the new ordering at price p0 must be strictly higher than expected profits under the original ordering at price p.
Proposition 5. Consider a simultaneous sales scheme, with all consumers in the same cohort. Suppose that before buying, consumers can simultaneously send a message m ∈ [θ, θ] × (0, λ) about their type which all other consumers observe, and where the seller observes M = (m1 , . . . , mn ) with probability q > 0. Furthermore suppose that the seller sets price p∗ if it does not observe M , and sets price p(M ) if it does, where p(M ) is non-decreasing P in ni=1 E(xi |p∗ , M ). Then when q is sufficiently small, an equilibrium exists where communication is informative, in the sense that each consumer truthfully reveals to all others the minimum level of total sales required for him to buy himself at price p∗ . In the limit as q tends to zero, consumer purchase decisions approach those in a setting where consumers all observe each others’ type, (θi , λi ) for all i = 1, . . . , n.
43
Proof. For consumer i, define Ni as the smallest value of N for which θi +
λi N − p∗ ≥ u0 . n−1
Ni is the minimum number of other consumers who must buy for consumer i to want to buy himself, given price p∗ . For each l = 0, 1, . . . , n − 1, let Bl denote the set of all (θ, λ) ∈ [θ, θ] × (0, λ) for which N = l. If n − 1 consumers buying is insufficient to motivate consumer i to buy, then we write Ni = n. Any consumer i with θi = θ has a strictly dominant strategy not to buy (Ni = n), regardless of the price. Let n0 denote the value of Ni for a consumer with θi = θ and λi = λ, where n0 ≤ n − 1 in any situation of interest. Notice that n0 = 0 if p∗ ∈ (θ − u0 , θ − u0 ), since then θi = θ implies a strictly dominant strategy to buy. Willingness to pay is increasing in θ which has full support on [θ, θ]. Hence, from an ex ante perspective, for each consumer i, there is a strictly positive probability that (θi , λi ) ∈ Bl , for each l = n0 , n0 + 1, . . . n. Consider a candidate equilibrium where each consumer i plays a mixed strategy placing strictly positive probability on all messages m ∈ BNi and zero probability on all m 6∈ BNi . Conditional on receiving any m ∈ BN from consumer i, all other consumers then infer that Ni = N . Notice that every m ∈ [θ, θ] × (0, λ) is on the equilibrium path, and corresponds to some N ∈ {n0 , n0 + 1, . . . , n}. Define Xl as the number of messages m ∈ Bl in this candidate equilibrium, for each P l = n0 , n0 +1, . . . , n. Define Nmax as be the maximum value of j +1 such that jl=0 Xl ≥ j +1; if no such j + 1 exists, then define Nmax ≡ 0. Then given price p∗ , consumer i’s strategy in this candidate equilibrium has him buy if and only if (Nmax − 1Imi ≤Nmax ) ≥ Ni . Thus, Nmax gives total sales at price p∗ , conditional on messages M = {m1 , . . . , mn }. Since all m ∈ [θ, θ] × (0, λ) are on the equilibrium path, there is a strictly positive probability that Nmax takes on each value 0, n0 + 1, n0 + 2, . . . , n. Given the messages of other consumers, any message mi ∈ BN leads to the same updated beliefs about consumer i’s type, (θi , λi ) ∈ BN , the same value of Nmax and the same purchase 44
behavior at price p∗ . Thus, any such message must also lead to the same price p and the same purchase behavior if the seller observes the messages. This is the case for N = n0 , n0 +1, . . . , n. It follows that for each N , consumers are indifferent between all messages m ∈ BN , so without loss of generality we can write m ∈ {n0 , n0 + 1, . . . , n). That is, each consumer i’s message is effectively an integer N , where the candidate equilibrium prescribes mi = Ni . The incentive to buy at a given price depends only on the number of other consumers expected to also buy. Hence, if the seller sets price p∗ , and each consumer i sends message mi ∈ Ni , then consumers will make the same purchase decisions as if they all observed each others’ type. To establish our result, we need to show that for q sufficiently close to zero, no consumer has a profitable deviation. First consider the case where the seller does not observe the messages so consumers face price p∗ . By mi = Ni for all i = 1, . . . , n and the definition of Nmax , each consumer who buys receives a payoff of at least u0 . Each consumer who does not buy would receive a payoff strictly less than u0 if he did buy. Hence, given price p∗ , a deviation from consumer i can only be profitable if it involves a change of message, to some m0i = Nk 6= Ni . Let Xl0 be the number of messages m = l following this deviation, for each l = n0 , n0 + 1, . . . , n. We have XN0 i = XNi − 1, XN0 k = XNk + 1, and Xl0 = Xl for all l 6= Ni , Nk . P 0 Define Nmax as the maximum value of j + 1 such that jl=0 Xl0 ≥ j + 1; if no such j + 1 0 ≡ 0. exists, then define Nmax
Suppose Nk > Ni , with Ni ≤ n − 1, so consumer i understates his willingness to P P pay. Then jl=0 Xl0 = jl=0 Xl for all j = 0, . . . , Ni−1 and for all j = Nk , . . . , n, whereas Pj Pj 0 0 0 0 ≤ Nmax − l=0 Xl = l=0 Xl − 1 for all j = Ni , . . . , Nk−1 . This implies Nmax − 1Imi ≤Nmax 1Imi ≤Nmax . Moreover, since all messages are on the equilibrium path, there is a strictly 0 0 positive probability that Nmax − 1Im0i ≤Nmax < Nmax − 1Imi ≤Nmax , for any realized value of
Nmax ∈ {0, n0 + 1, n0 + 2, . . . , n}. The payoff of reporting Ni is given by E(Nmax ) − 1 (1−q) P(Nmax − 1Imi ≤Nmax < Ni )u0 + P(Nmax − 1Imi ≤Nmax ≥ Ni ) θi + λi +qU0 , n−1 where it understood that the term E(Nmax ) is conditional on P(Nmax − 1Imi ≤Nmax ≥ Ni ), and 45
where U0 is the expected payoff if the seller observes the messages. Meanwhile the payoff from deviating to Nk is 0 )−1 E(Nmax 0 0 0 0 +qU1 , ≥ Ni ) θi + λi < Ni )u0 + P(Nmax − 1Im0i ≤Nmax (1−q) P(Nmax − 1Im0i ≤Nmax n−1 0 0 0 ≥ Ni ), and − 1Im0i ≤Nmax ) is conditional on P(Nmax where it understood that the term E(Nmax
where U1 is the expected payoff obtained by consumer i if the seller observes these messages which include m0i . Taking the difference between the two payoffs and using the fact that 0 ), the deviation is not profitable if: E(Nmax ) > E(Nmax 0 E(Nmax )−1 q ∆P θi + λi − u0 > (U1 − U0 ), n−1 1−q 0 0 − 1Im0i ≤Nmax ≥ Ni ) > 0. This inequality where ∆P ≡ P(Nmax − 1Imi ≤Nmax ≥ Ni ) − P(Nmax
holds for sufficiently small q, thus underreporting willingness to pay is not profitable. Now suppose Nk < Ni , with Ni ≥ 1, so consumer i overstates his willingness to pay. Again consider the case where the seller does not observe messages, so consumers face price P P 0 0 ≥ Nmax holds, but Nmax > Nmax p∗ . Then jl=0 Xl0 = jl=0 Xl for all j ≥ Ni . Hence, Nmax 0 0 > Nmax is necessary for the deviation < Ni . The condition Nmax can only hold if Nmax
to increase consumer i’s payoff, since the number of consumers other than i who buy must 0 increase. But Nmax < Ni implies that consumer i will not buy himself following the deviation,
so the deviation will not increase his payoff. Continue to suppose Nk < Ni but consider the case where the seller does observe 0 messages. Then Nmax ≥ Nmax implies that the deviation leads to a weakly higher price:
p(M 0 ) ≥ p(M ), where M denotes the equilibrium messages, and M 0 denotes messages given the deviation. From (2), consumer best-response functions when simultaneously making purchase decisions are upward-sloping (strategic complements), where a price increase reduces the net payoff from buying. Thus, p(M 0 ) ≥ p(M ) implies E(xj |M 0 ) ≤ E(xj |M ) for each consumer j (see Vives (1990)), so the deviation will not increase consumer i’s payoff. Hence, overreporting willingness to pay is not profitable. 46
Finally, note that as q approaches zero, the probability that the seller charges p∗ approaches 1, which together with informative communication guarantees that consumers almost surely make the same purchase decisions as if they observed each others’ types.
Proposition 5.
Suppose the seller can commit to a dynamic pricing schedule, with price
p(Kt ) for cohort t conditional on previous sales Kt . Then fully sequential sales (a single consumer per cohort), delivers higher expected profits than fully simultaneous sales (all consumers in a single cohort). Proof. First consider a fully simultaneous scheme, with all consumers in a single cohort: I = {I1 }, with n1 = n. Then dynamic pricing is equivalent to static pricing; both simply specify a single value of p. Let π(p∗ |I) denote expected profits given the optimal static price p∗ under this partition. Now consider a fully sequential scheme, with a single consumer per cohort: I’ = {I10 , . . . , In0 }, with nt = 1 for all t = 1, . . . , n. For each t, the seller’s strategy specifies a price p(Kt ), for every possible value of previous sales Kt = 0, . . . , t − 1. With slight abuse of notation, let π(p(Kt )|I’) denote expected profits given this pricing schedule under this partition. Suppose that for each t = 1, . . . , n, the seller sets p(Kt ) = p∗ for all Kt = 0, . . . , t − 1. Expected profits are then π(p∗ |I’). Let π(p(Kt )∗ |I’) denote expected profits given the optimal dynamic pricing schedule p(Kt )∗ . Then optimality implies π(p(Kt )∗ |I’) ≥ π(p∗ |I’). Corollary 1 shows that π(p∗ |I’) > π(p∗ |I), which in turn implies π(p(Kt )∗ |I’) > π(p∗ |I).
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